Grade 6 Workbook — Chapter 14: Circles: Circumference and Area

SOL 6.MG.1 · Companion to Textbook Chapter 14

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/.

PAGE 1 — Chapter opener

Chapter 14 · Circles: Circumference and Area

Standard 6.MG.1

In this chapter you will:

• Identify chord, diameter, radius, circumference, and area
• Investigate how radius, diameter, and circumference are related
• Gather data to discover pi
• Build and use the circumference formula
• Solve circle problems, including real-world ones

Words to know: circle · center · radius · diameter · chord · circumference · area · pi · semicircle

Formula bar: $d = 2r$ · $r = \dfrac{d}{2}$ · $C = \pi d$ · $C = 2\pi r$ · $A = \pi r^2$ · use $\pi \approx 3.14$

PAGE 2 — Parts of a circle

14.1 Parts of a Circle

FIGURE: fig1-circle-parts.png (full width) [Circle with center O; labeled radius OA, diameter BC through the center, chord DE not through the center, an arrow to the curve labeled circumference, and the shaded interior labeled area.]

Label each part on the figure above, then complete the definitions.

A radius goes from the ____________ to a point on the circle.

A diameter passes through the ____________ and has both endpoints on the circle.

A chord has both endpoints ____________ , and does not have to pass through the center.

Circumference is the distance ____________ the circle. Units: ____________

Area is the surface ____________ the circle. Units: ____________

True or false.

PAGE 3 — Radius and diameter

$d = 2r$ and $r = \dfrac{d}{2}$

FIGURE: fig2-radius-diameter.png (right half of page) [Two circles: one labeled r = 5 cm with diameter 10 cm and a bracket showing two radii; one labeled d = 14 in with radius 7 in.]

Complete the table.

Apply it. A round pond measures $40$ ft straight across through the middle.

That measurement is called the ____________ . Distance from center to edge: ______ ft, called the ____________ .

Explain. Why can a chord never be longer than the diameter?

PAGE 4 — Exit ticket 14.1

Exit Ticket · Lesson 14.1

Name: ________________________ Date: ____________

1. A circle has radius $11$ cm. Diameter: ______

2. A circle has diameter $9$ in. Radius: ______

3. Define chord in your own words.

4. Why is circumference measured in units but area in square units?

PAGE 5 — Wrap the string

14.2 Discovering Pi

FIGURE: fig4-wrap-string.png (full width) [A can lid with string wrapped once around it; below, the string laid straight with three diameter-lengths laid end to end along it and a short piece left over.]

The big question: How many diameters fit around a circle?

Your answer after wrapping the string: about ______ diameters and a little more.

Measure three round objects. Record and compute.

Average of your three ratios: ______

The number you are closing in on is called ____________ , written $\pi$, and we use $\pi \approx$ ______ .

PAGE 6 — Class data set

Comparing $C$ to $d$

Compute each ratio to the nearest hundredth.

FIGURE: fig5-pi-scatter.png (half width) [Scatter plot of diameter versus circumference for five circles with a straight line through the origin labeled C = 3.14d.]

What do all seven ratios have in common?

Explain. Why does almost nobody measure exactly $3.14$?

PAGE 7 — Exit ticket 14.2

Exit Ticket · Lesson 14.2

Name: ________________________ Date: ____________

1. $C = 18.9$ cm and $d = 6.0$ cm. Find $C \div d$ to the nearest hundredth. ______

2. What number does $C \div d$ approach for every circle? ______ Its name: ____________

3. About how many diameters fit around a circle? ______

4. Why are measured ratios not all the same?

PAGE 8 — Building the circumference formula

14.3 Circumference

FIGURE: fig6-circumference-formula.png (full width) [Two circles: one with diameter labeled d and C = pi times d beneath; one with radius labeled r and C = 2 times pi times r beneath; a two-way arrow labeled d = 2r between them.]

Fill in the derivation.

$$\pi = \frac{C}{\_\_\_}\qquad \Longrightarrow \qquad C = \pi \times \_\_\_$$

Because $d = 2r$, replacing $d$ gives $C = \pi(\_\_\_) = \_\_\_\pi r$.

Which formula would you use?

Find each circumference. Show the substitution.

1. $d = 10$ cm → $C = 3.14 \times$ ______ $=$ ______

2. $d = 20$ in → ______

3. $r = 4$ ft → $C = 2 \times 3.14 \times$ ______ $=$ ______

4. $r = 9$ m → ______

PAGE 9 — Circumference practice

Circumference Practice

Find each circumference. Include units.

Work backward.

5. $C = 94.2$ ft. Diameter: ______ Radius: ______

6. $C = 31.4$ m. Diameter: ______

7. $C = 18.84$ cm. Radius: ______

Doubling. Circle A: $r = 3$ cm → $C =$ ______ Circle B: $r = 6$ cm → $C =$ ______

What happened to the circumference when the radius doubled? _______________

Error hunt. Priya wrote: "$r = 8$ cm, so $C = 3.14 \times 8 = 25.12$ cm."

Her error: _______________________________________________

Correct answer: ______

PAGE 10 — Exit ticket 14.3

Exit Ticket · Lesson 14.3

Name: ________________________ Date: ____________

1. $d = 15$ cm → $C =$ ______

2. $r = 5$ in → $C =$ ______

3. $C = 31.4$ m → $d =$ ______

4. Why do $C = \pi d$ and $C = 2\pi r$ always agree for the same circle?

PAGE 11 — Where the area formula comes from

14.4 Area of a Circle

FIGURE: fig7-area-sectors.png (full width) [A circle cut into 8 equal wedges, then the wedges rearranged into a row alternating point-up and point-down to form a near-parallelogram; the slanted height labeled r and the base labeled half the circumference, pi times r.]

Complete the reasoning.

The height of the near-parallelogram is the ____________ , or $r$.

The base is half the circumference, which is $\dfrac{2\pi r}{2} =$ ______ .

Area of a parallelogram $=$ base $\times$ height, so $A = (\pi r)(r) =$ ______ .

Order matters. In $A = \pi r^2$, square the ____________ first, then multiply by $\pi$.

For $r = 5$: $5^2 =$ ______ , then $3.14 \times$ ______ $=$ ______

FIGURE: fig8-area-grid.png (half width) [A circle of radius 5 on a square grid with whole interior squares shaded darker and partial edge squares shaded lighter.]

Counting squares gives about ______ square units, which matches the formula.

PAGE 12 — Area practice

Area Practice

Find each area. Include square units.

Work backward.

1. $A = 50.24\ \text{ft}^2$. Radius: ______

2. $A = 28.26\ \text{m}^2$. Radius: ______

Same circle, two measurements. $r = 6$ cm.

$C =$ ______ $A =$ ______ Why are the units different? _______________

Error hunt. Devon wrote: "$d = 10$ cm, so $A = 3.14 \times 10^2 = 314\ \text{cm}^2$."

Error: _______________________________________________

Correct area: ______ Devon's answer was ______ times too large.

PAGE 13 — Exit ticket 14.4

Exit Ticket · Lesson 14.4

Name: ________________________ Date: ____________

1. $r = 9$ cm → $A =$ ______

2. $d = 4$ in → $A =$ ______

3. $A = 28.26\ \text{m}^2$ → $r =$ ______

4. Why is area reported in square units but circumference is not?

PAGE 14 — Which measurement does the job need?

14.5 Circle Problems in Context

FIGURE: fig9-sprinkler.png (right half of page) [Overhead view of a lawn with a sprinkler at the center of a shaded circle of watered grass, radius labeled 12 ft.]

Circle the measurement each job needs.

Then ask: were you given the radius or the diameter?

"Reaches $12$ ft in every direction" gives the ____________ .

"A pizza is $16$ inches" gives the ____________ .

Solve.

1. Sprinkler, $r = 12$ ft. Area watered: ______

2. Pen $40$ ft across. Fence needed: ______

3. Round rug, $r = 4$ ft. Area: ______

4. Trampoline $14$ ft across. Edge padding: ______

PAGE 15 — Real-world circles

Circle Problems

Show your work in the space beside each item.

5. A pool is $18$ ft across. Distance around: ______ Surface area: ______

6. A wagon wheel is $2$ ft across and turns $100$ times. Distance rolled: ______

7. A garden has $r = 10$ ft. Fencing costs $4 per foot. Total cost: ______

8. A pizza pan has $r = 7$ in. Area of the pan surface: ______

9. Two $10$-inch pizzas or one $16$-inch pizza? (Sizes are diameters.)

   Two $10$-in: ______ One $16$-in: ______ More pizza: ______ By how much: ______

10. A round table is $5$ ft across. A cloth must hang $6$ in past the edge all around.

    Cloth diameter: ______ Fabric area: ______

11. Semicircle. A window is a $4$ ft by $3$ ft rectangle with a semicircle on top.

FIGURE: fig10-semicircle-window.png (half width) [Window shaped as a rectangle 4 ft wide and 3 ft tall with a semicircle of radius 2 ft on top, straight edge matching the rectangle's width.]

Rectangle area: ______  Semicircle area: ______  Total glass: ______

12. Doubling. Complete the table, then explain.

Doubling the diameter multiplies $C$ by ______ and $A$ by ______ .

Why the difference? _______________________________________________

PAGE 16 — Exit ticket 14.5

Exit Ticket · Lesson 14.5

Name: ________________________ Date: ____________

1. A tabletop is $6$ ft across. Trim needed: ______

2. A sprinkler reaches $9$ ft in every direction. Area watered: ______

3. A wheel $3$ ft across makes $50$ turns. Distance: ______

4. To find how much sod a circular lawn needs, which formula do you use, and why?

PAGE 17 — Chapter 14 review, part 1

Chapter 14 Review

Use $\pi \approx 3.14$. Include units.

Part A · Parts of a circle

1. Name each: a) segment from center to circle ____________ b) segment with both endpoints on the circle ____________ c) distance around ____________ d) surface inside ____________

2. Difference between a chord and a diameter: _______________________________________________

3. Which uses square units, $C$ or $A$? ______ Why? _______________

4. A circle has $r = 15$ cm. Length of its longest chord: ______

Part B · Radius, diameter, circumference

5. Diameter: a) $r = 8$ in ______ b) $r = 3.5$ ft ______

6. Radius: a) $d = 50$ m ______ b) $d = 9$ cm ______

7. $r = 2$ cm → $C =$ ______ ; $r = 6$ cm → $C =$ ______ ; effect of tripling: _______________

8. The circumference of a circle is always about ______ times its diameter.

Part C · Approximating pi

9. $C \div d$ to hundredths: a) $C = 25.1$ cm, $d = 8.0$ cm ______ b) $C = 44.0$ cm, $d = 14.0$ cm ______

10. Average of 9a and 9b: ______ This approximates ______ .

11. A group reports $3.55$. Likely cause: _______________________________________________

PAGE 18 — Chapter 14 review, part 2

Chapter 14 Review (continued)

Part D · The circumference formula

12. From $\pi = \dfrac{C}{d}$, show how to get $C = \pi d$: _______________________________________________

13. Why does $C = 2\pi r$ follow from $C = \pi d$? _______________________________________________

14. Circumference: a) $d = 22$ cm ______ b) $r = 15$ in ______

15. $C = 43.96$ ft. Diameter: ______ Radius: ______

Part E · Problems with circumference and area

16. Area: a) $r = 12$ cm ______ b) $d = 26$ in ______

17. A fountain has $r = 6$ ft. Distance around: ______ Surface area: ______

18. A tire is $30$ in across. One turn: ______ Twenty turns: ______

19. A patio is $20$ ft across. Concrete $5 per square foot, border $8 per foot.

    Concrete cost: ______ Border cost: ______ Total: ______

20. Radii $5$ cm and $10$ cm. Show with numbers why doubling the radius doubles $C$ but multiplies $A$ by four.

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Canva production notes

• Page size: 8.5 × 11 in, 0.75 in margins
• Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
• Formula bar: repeat the page 1 formula bar as a narrow footer strip on pages 8–16 so students are not flipping back
• Figure widths: fig1-circle-parts.png, fig4-wrap-string.png, fig6-circumference-formula.png, and fig7-area-sectors.png full width; the rest half width
• Measuring page: page 5 needs a blank three-row table students fill with their own objects; leave the rows tall enough to write in pen
• Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow
• Work space: pages 9, 12, and 15 need a right-hand column at least 2 in wide for shown work